Answer to Question #123401 in Abstract Algebra for Aditya Karn

Question

Any two groups of order m are isomorphic, where m ∈ N . True or False. justify

Answer 1

Let’s consider two groups $Z_4$ and $U(8)$

$Z_4=\{0,1,2,3\}$ addition $\mod 4$

$U(8)=\{1,3,5,7\}$ multiplication $\mod 8$

$|Z_4|=|U(8)|=4$

Suppose, that there exists isomorphism $f: Z_4\rightarrow U(8)$ .

Then $f$ will map the identity element of $Z_4$  to the identity element of $U(8)$ : $f(0)=1$ .

$f(1)$ can be equal to $3,5,7.$

$f(2)=f(1+1)=f(1)\times f(1)=1,$ because $3^2=5^2=7^2=1\mod 8$ .

We have, that $f(2)=1,$ but $2$ isn’t identity element of $Z_4$ .

Contradiction.

So, there is no isomorphism between $Z_4$ and $U(8)$ .

Although both these groups have order 4, they aren’t isomorphic.

Answer: False.